INTERSECTION AND COMBINATION OF DIGITAL ELEVATION MODELS - METHODS AND APPLICATIONS
Manfred Sigle, Olaf Hellwich, Andre Kostli
INPHOGmbH
Smaragdweg 1
7000 Stuttgart 1, Gennany
Commission IV
ABSTRACT:
Three-dimensional surface descriptions, which exist in the form of digital elevation models, are well suited - by using any
mathematical function - for a combination of different types of surface data (DEM, digital slope model, digital soil value model,
etc.), as well as for the intersection of these digital surface models with polygon areas. Corresponding methods were realized in the
DEM program system SCOP.
Typical application examples are the derivation of height difference models from two DEMs, a subsequent volume computation, the
determination of slope statistics from a digital slope model, or the production of soil loss prediction maps by combining the
influence factors for soil erosion.
KEY WORDS: DEM, volume computation, soil loss prediction, polygon overlays
INTRODUCTION
SCOP has been used world-wide by numerous organizations
for all types of DEM projects, from large-scale engineering
projects up to the generation of national DEMs. SCOP is a
joint development by INPHO GmbH (Prof. Ackermann) and
by the Photogrammetric Institute of the Technical University
Vienna (Prof. Kraus). The structure of SCOP is shown in
fig. 1.
In addition to the conventional DEM applications (isolines,
profiles, perspective views) arithmetic operations on the DEM
data like the DEM intersection and the functional combination
of DEM data represent a new challenge for DEM systems.
DEM intersection and combination methods are already used
for volume computation. In addition, they open many new
fields of application to a DEM system.
Output
DTM applications
Up to now surface data describing the terrain (e.g. elevation,
slope) have usually been intersected with polygon areas by
polygon overlays. In a first step the surface data had to be
classified and described by polygon areas with constant class
values (e.g. slope class areas derived from a digital slope
model). This classification includes a considerable loss of
information which leads to erroneous results, especially in
cases where the polygon data are used for a functional
combination with other terrain data (e.g. for soil loss
prediction).
SCOP .ISOLINES
v ..ctor
graphics
SCOP.PROFILE
ro,ster
SCOP.SLOPE
gro.phlcs
I
I
SCOP,PERSPECT
SCOP.INTERSECT
I
I
Therefore methods had to be developed for intersection and
combination which made full use of the three-dimensional
surface information of aDEM.
SCOP.PIXEL
Together with the necessity of having a SCOP module for
volume computation, the above considerations were the reason
for the development of the program SCOP.INTERSECT.
------
CiIS-
Interfo.ces
nUl'lerlCo.l
output
SCOP.PLOT
I
DTH accesS routines
SCOP.CONVERT
I .......
other
systeMS
Fig. 1: SCOP modules
2. THE DEM PROGRAM SYSTEM SCOP
SCOP is a multi-purpose program system for the generation
and application of digital elevation models (Kraus et al. 1982,
Ackermann 1991). Its flexible high-quality interpolation
methods produce DEMs of cartographic quality. SCOP
includes all common DEM application modules which derive
follow-up products like single heights, isolines, profiles,
volumes, perspective views, digital slope models and raster
graphics DEM representations.
3. DATA STRUCTURE OF THE SCOP DEM
The heart of a DEM system is the digital elevation model
which is characterized by the quality of its interpolation
methods and by the efficiency of its data structure.
878
SCOP interpolates a highly densified grid model with variable
grid width (Kostli/Wild 1984) from irregularly distributed
points (mass points, spot heights) and lines (break lines, form
lines, border lines) of any origin. The measured lines are
rigorously stored in the DEM and are therefore available for all
DEM applications. A subdivision of the DEM area into small
rectangular computing units enables the processing of huge
DEM projects even on personal computers. The data structure
is shown in fig. 2.
The DEM is stored as an index-sequential file with direct
access to local parts (Kostli/Sigle 1986). Thus a very efficient
processing of the DEM data is guaranteed, with access times of
less than 0.1 sec. even in an extremely large DEM.
DEMI
DEM2
difference model
difference isolines
area no.
The SCOP DEM data structure is not exclusively used for the
storage of terrain heights, but it is also the data basis for
derived data like digital slope models or height difference
models. Those digital surface descriptions are called "SCOP
models" in the following.
2
areas of interest
~
-
cutting:
filling:
cutting:
filling :
cutting:
filling:
area
volume
25402.48
921.57
1357.36
0.00
2219.11
\89.90
144809.41
204.08
15569.44
0.00
6358.70
107.49
results of volume computation
Fig. 3: Example of a volume computation
II.
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i"--f\
4.2 Volume Computation
/'
It is the aim of a volume computation to determine the earth
"n
volumes separated into cutting and filling for some areas of
interest (e.g. earthwork areas).
II
Areas of interest may be any areas within the difference model
defined by closed polygons. Cutting and filling are separated
by the intersection line of the two DEMs, which is the isoline
of height difference zero.
./
Fig.2: DEM data structure
.
...,....-
/I
4. VOLUME COMPUTATION - A FIRST EXAMPLE
FOR DEM COMBINATION AND INTERSECTION
I
I
I
'-.
.-.-.- -.-.-
- grid of the difference model
border of the area of interest
I
I
I
A volume computation from digital elevation models may be
done in two steps:
1. derivation of a height difference model
2. volume computation from the difference model for
predefined areas of interest (e.g. earthwork areas).
intersection line
(height difference zero)
break lines
FigA.l: Lines for volume
computation
4.1 Derivation of a Height Difference Model
Two DEMs for the same area have to be given in the SCOP
DEM data structure. They may describe a former or existing
terrain, but also projected terrain forms.
The two DEMs may have different grid intervals and different
line information (break lines etc.). From the two DEMs the
height differences are computed and stored in the difference
model as shown in fig. 3 for an open mining area.
The difference model inherits the grid structure from one of the
DEMs and the line information from both DEMs. Thanks to
this, embankments and other important terrain features are
fully represented in the difference modeL
FigA.2: Triangular prisms
879
Volume computation has to start from a grid of height
differences and from different line information' (limit of the
area of interest, intersection line, break lines, form lines, DEM
border lines). Fig. 4.1 shows the initial lines for a small part of
a difference model.
The lines build up irregular polygons. For each grid mesh a
triangular network is derived from the irregular polygons (see
fig. 4.2). Thus, the volume computation can be reduced to
triangular prisms which have to be related to either cutting or
filling, depending on the sign of the height differences.
5.2 DEM Intersection
Volume computation is a special case of DEM intersection for
which the DEM is a difference model. It is intersected with
areas of interest. Each area of interest is subdivided by the
isolines of height difference zero into areas of cutting and
filling. The intersection results are volumes of cutting and
filling.
5. GENERAL VIEW OF THE COMBINATION
AND INTERSECTION METHODS
The methods used for volume computation shall in the
following be used as an example for a more general view of
the arithmetic operations on DEM data.
In general the DEM can be any SCOP model (DEM, slope
model or any functional model). The areas of interest may be
any polygon areas. And the height difference zero for dividing
cutting and filling may be replaced by any class limits.
5.1 DEM combination
The intersection classifies the polygon area and computes the
intersection results which may be classified areas, volumes or
surfaces. Applications of a DEM intersection are described in
chapter 7 and in table 1.
Building up a difference model from two DEMs is a functional
DEM computation by using a simple subtraction as a
combination function.
Zdiff. = ZDEMI
5.3 Polygon Overlays
- ZDEM2
In many cases the input data for a functional combination of
surface data or for an intersection do not exist in form of a
DEM, but have to be digitized from maps in form of polygon
areas. Each polygon area has a corresponding value Z.
SCOP.INTERSECT is now extended for the use of any
mathematical function of the form
Such
polygon
areas
may
be
converted
with
SCOP.INTERSECT into a SCOP model, and are then available
for a functional combination with other surface data or for an
intersection.
The function f is described by the fundamental arithmetic
operations and by a discrete description of more complex
functions.
The functional values Zfunct. are stored in a SCOP model with
the grid structure of either the Zl or the Z2 SCOP model and
the line information of both models.
,-
r-.-e. ....
"
......
/'
U
P
II
II
'fl
II
I'
1-1-1
-
.......
SCOP model 1
SCOPmode12
I
I
./
1/
Ll
k::i>
II
'-
After a conversion of polygon areas into a SCOP model
SCOP.INTERSECT is able to solve a conventional polygon
overlay by intersecting a second set of polygon areas with the
SCOP modeL
1/
,
1\..1/
,\::::::1:10>
+ .....
The conversion into a SCOP model is done by overlaying the
polygon areas with a regular grid. Each grid and polygon point
is stored with the Z value of the referring polygon area. The
result is a surface description which consists of adjacent
horizontal terraces.
r-t--rt--.
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_/,..~ f')~
Each point of the functional model needs its values Zl and Z2
before performing the computation of the functional Z value.
In most cases only one of the Z values is directly stored either
in the Zl or in the Z2 model. The corresponding Z value of the
other model is automatically determined by an interpolation
within the respective grid mesh.
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6. OTHER APPLICATIONS FOR ADEM
COMBINATION
.....
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The two following examples show how other arithmetic or
boolean operations can be used for a combination of digital
elevation models.
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6.1 Integration of planned structures into aDEM
"Ii.:t::
functional model
A simple boolean function can be used for the integration of
planned terrain surfaces (roads, railways, embankments etc.)
into a DEM of the existing terrain.
Fig. 5: Data structure of the functional model
880
Input data are two DEMs, one for the existing terrain, and the
other for the planned structures surrounded by border lines. In
the combined DEM the planned elevations are stored where the
planning DEM exists. Outside the planned structures the Z
values of the existing terrain are taken.
The combined DEM can then be represented in form of
perspective views or evaluated in form of profiles.
6.2 Prediction of Soil Erosion
A more complex function is used for the combination of the
influencing factors for soil erosion. In most investigations the
expected annual soil loss per unit area is described by the
universal soil loss equation (USLE) (WischmeierlSmith 1978)
as the product of 6 influencing factors:
K-factors
A = R*K*L*S*C*P
Slope-length classes
with
A = expected annual soil loss per unit area
R = rainfall factor
K = soil erodability factor
L = slope-length factor
S = slope-steepness factor
C = cover and management factor
P = support practice factor
...
< 50m
50 -120 m
120 - 200 m
I##l
Amongst other applications, the USLE is being applied to
terrain planning in land consolidation projects (Sigle 1991).
The maximum tolerable slope length of the restructured terrain
is computed by combining the slope factor S (derived from a
digital slope model) and the soil erodability factor K. K is
stored in a SCOP model which was built up from digitized
polygon areas of a map of soil classes. All other influence
factors could be kept constant for a local land consolidation
area.
> 200 m
tolerable slope-length
Fig. 6: Soil loss prediction for a land consolidation project
7. OTHER APPLICATIONS FOR ADEM
INTERSECTION
Two applications are described in the following which are used
for the estimation of agricultural land (e.g. for land
consolidation projects).
The combination results in a digital slope-length model which
can be represented in a map of classes of tolerable slopelength.
7.1 Slope Statistics
A graphical presentation is given for a small part of the land
consolidation project Sulzfeld in fig. 6. Data acquisition was
done by the land consolidation authority of BadenWlirttemberg. In practical use the soil loss prediction maps
could be co~siderably improved compared to fig. 6 by using
colour hatchmg or a z-coded raster representation for the slopelength classes.
SCOP includes a module for the derivation of a digital slope
model from a DEM. The slope model has the same data
structure as the DEM, but the terrain heights are replaced by
slope values (steepness in per cent). Break line information is
rigorously considered in the slope model.
For land estimation the slope model is intersected with pieces
of land (polygon areas) by using several slope values as class
limits. The results are the slope class areas for each piece of
land.
The Sulzfeld project has an extension of 5km x 5km. The
DEM data were acquired by a photogrammetric grid
measurement (44 178 points including 18461 points on break
lines, form lines and border lines). The K-factors were
digitized by 869 polygon areas with 16485 polygon points.
7.2 Soil Value Statistics (Polygon Overlay)
An example for a polygon overlay is the computation of soil
value areas for pieces of land.
The total project was realized under MS-DOS on a 80386 PC
(?3 MHz). It re~uired a disk capacity of 40 MB and computing
tlmes of 64 lllnutes for DEM interpolation, 18 minutes for
derivation of a digital slope model, 9 minutes for building up
the K factor model, 58 minutes for the derivation of the slopelength model by combining the two SCOP models and 3
minutes for the output of a soil loss prediction map for ~he total
area.
Soil value classes usually exist in form of polygon areas in soil
value maps. The polygon areas have to be digitized and to be
converted into a SCOP model which is then intersected with
another set of polygon areas (e.g. pieces of land). Class limits
for the intersection may be any soil values in between the
different soil value classes of the map.
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8. SUMMARY
REFERENCES:
Flexible, accurate, fast and rigorous methods were developed
on the basis of the SCOP DEM data structure in order to
combine and to intersect different types of surface data. The
methods were made available for a practical use by the
program SCOP.INTERSECT.
Ackermann,F., 1991. High-quality digital terrain models - The
SCOP program and derived products. Proceedings of the
EARSEL workshop on relationship of remote sensing and
geographic information systems, Hannover.
Kostli,A., Sigle,M., 1986. The random access data structure of
the DTM program SCOP. Int. Arch. Photogramm. Remote
Sensing, vo1.26, part 4, pp.45-52, Edinburgh.
In addition to the applications described in this paper there are
many similar other ones possible. They cover a wide field
ranging from projects which are closely connected with digital
elevation models to projects which only make use of the DEM
data structure as a data basis for polygon overlays.
Kostli,A., Wild,E., 1984, Digital elevation model featuring
varying grid size.
Int. Arch. Photogramm. Remote
Sensing, vol.25, part 3, Rio de Janeiro.
A summary of the methods and applications is given in table 1.
Kraus ,K. , Assmus,E., Kostli,A., Molnar,L., Wild,E., 1982,
Digital elevation models: users' aspects. Proc. 38 th Photogrammetric Week 1981, pp.165-181, Stuttgart.
This example shows that a DEM program can no longer be
seen as a stand-alone system. Today's methods are carefully
designed to be of general use, e.g. in geographic information
systems.
Sigle,M., 1991, Die Erstellung von Bodenerosionsgefahrdungskarten auf der Basis eines digitalen GeHindemodells.
GIS, Jahrgang 4, Heft 4/1991, pp.2-7, Karlsruhe.
Wischmeier,W.H., Smith,D.D., 1978,
Predicting rainfall
erosion losses - a guide to conservation planning. USDA
Agricultural Handbook No. 37, Washington.
Application:
volume
computation
slope statistics
integration of
planned terrain
soil loss
prediction
polygon overlays
1. DEM combination
input model 1
DEMl
DEM (existing)
slope model
input model 2
DEM2
DEM (planned)
erodability model
function
subtraction
boolean function
USLE
-!--
-.it
-.it
functional model
difference model
combined DEM
slope-length model
follow-up products
difference maps
perspective views
soil erosion maps
2. DEM intersection
SCOPmodel
difference model
slope model
soil value model
polygon areas
earthwork areas
pieces of land
pieces of land
class limits
height difference 0
slope classes
soil value classes
intersection result
-.¥
volumes for
cutting and fIlling
areas of slope
classes
-.it
-!--
areas of soil
value classes
Table 1: Summary of methods and applications
882